Is it better to offer a missed Fajr prayer before Dhuhr prayer, or does it make no difference in the makeup?

There is no sin upon you both if you miss Fajr prayer after taking the means to wake up for it. Otherwise, sin is upon you both, and you must hasten to repent and seek forgiveness. The missed Fajr prayer must be made up immediately, even if it is during a time when prayer is prohibited.